_s u2 ds in .NET Implementation Code 39 Full ASCII in .NET _s u2 ds

_s u2 ds using barcode integrating for visual .net control to generate, create 3 of 9 barcode image in visual .net applications. GTIN-8 r 2 d 9 ! = E 2 " 39 barcode for .NET "2 2"11 "22 G"2 ds 22 12 ; 1  2 11 ! @ 2 gz r r d @t. (9:90a). 8 <1 Z Fj rj  :2. (9:90b).  div r d  (9:90c). where s is the shell mat barcode code39 for .NET erial volume, us is the middle surface displacement of the shell in bending, E is Young s modulus of elasticity, "ij are strain components expressed in terms of the shell middle surface displacements, G is the modulus of rigidity. Carrying out the variational of equation (9.

89) gives the equations of motion and the boundary conditions. The boundary value problem of the liquid is given in terms of the velocity potential function in the form r2 0 @ un @n @ _ w @r inside the liquid domain on the liquid free surface (9:91a) (9:91b). at the wetted shell surface (9:91c). @ 1 r 2 gz 0 @t 2 on the liquid free surface (9:91d). where n is a unit vector VS .NET 3 of 9 along the outward normal of the free surface, un is the normal component of the liquid relative velocity with respect to the free surface, w is the shell deflection at the wetted surface. Boyarshina (1984) showed that in the elastic shell liquid system a rotational motion of the free surface of the liquid leads to the excitation of the elastic shell in the form of traveling waves.

9.5 Nonlinear interaction with nonlinear sloshing in the azimuthal directio Code39 for .NET n. The rotational motion and the associated traveling waves owe their origin to the nonlinear coupling between the coordinates describing the liquid orthogonal modes and the elastic vibrations of the shell.

The interaction between these modes in the neighborhood of an internal resonance condition will be considered. Let the perturbed free surface of the liquid be represented by the two-parameter expansion XX  r; ; t asi t cos i is t Rsi r (9:92). i 0 s 1 where asi (t) are the gen Code 39 Full ASCII for .NET eralized coordinates, is (t) are the angular coordinates, and Rsi (r) are the eigenfunctions of the homogeneous boundary-value problem   d 2 Rsi r 1 dRsi r i2 2 si 2 Rsi r 0 (9:93a) dr2 r dr r  dRsi r   0 dr r R (9:93b). where si is the sth eige nvalues, or the roots of the resulting Bessel function that satisfies condition (9.93b). For the case of the ends of a simply supported shell, the shell deflection may be approximated by the wave expansion w z; ; t XX.

bmn t cos n mn t sin mp z h =l (9:94). where bmn (t) are the generalized coordinates and mn (t) are the angular co .NET bar code 39 ordinates of the shell deflection, m 1, 2, 3, . .

., and n 2, 3, 4, . .

. The liquid velocity potential function may be written as the sum of two components l s (9:95). where l is due to the fr ee-surface wave motion of the liquid, and s is due to the elastic deformation of the shell. These two components can be evaluated by using equations (9.92) and (9.

94), and the following expressions are obtained X X _ _ l ai i Ai0 sin i i ai Ai0 cos i i . _ ai aj Aij cos i i Code39 for .NET cos i j _ ai aj ak i Aij sin i i cos j j cos k k (9:96a). XX X i j k for which i, j, k 0,1,2, . . . The subscript s has been dropped. s XX X m b k h i _ _ Bmnk bmn cos n mn bmn mn sin n mn (9:96b). Nonlinear interaction under external and parametric excitations where Ai0, Aij, and Aijk are harmonic functions depending on the spatial coordinates and they must satisfy the boundary condition of the liquid free surface. The functions Bmnk are evaluated from the shell boundary-value problem and are given by the expression 2 In pkr=h pk Bmnk z h cos 0 pkR=h np In h ! Zh w z; ; t cos. ! pk z dz h (9:97). A set of nonlinear, coupl Code39 for .NET ed, ordinary differential equations describing the time evolution of the generalized coordinates amn (t) and bmn (t) can be obtained by applying Gaerkin s method, or another discrediting approach (Narimanov, 1957b and Narimanov, et al., 1977), to the dynamical boundary condition (9.

91d) and using equations (9.95) to (9.97), gives   2 _ mn 1 D 1 amn mn 2 amn C 1 bmn cos .

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